On the Narasimhan-Seshadri correspondence for Real and Quaternionic vector bundles

TL;DR

This paper extends the Narasimhan-Seshadri correspondence to Real and Quaternionic vector bundles over Klein surfaces, establishing a bijection between semi-stable bundles and certain orbifold group representations.

Contribution

It introduces an invariant-theoretic approach to relate semi-stable Real and Quaternionic bundles with orbifold fundamental group representations, generalizing classical results.

Findings

Unique unitary orbit of projectively flat, Galois-invariant connections exists within the closure of each semi-stable orbit.

S-equivalence classes of semi-stable bundles correspond bijectively to representations of orbifold fundamental groups.

The approach provides a new perspective on the moduli space structure for Real and Quaternionic bundles.

Abstract

Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then use this invariant-theoretic perspective to prove a version of the Narasimhan-Seshadri correspondence in this context: S-equivalence classes of semi-stable Real and Quaternionic vector bundes are in bijective correspondence with equivalence classes of certain appropriate representations of orbifold fundamental groups of Real Seifert manifolds over the Klein surface M.